Simplify and expand the following expression: $ \dfrac{2}{2p + 18}+ \dfrac{2}{2p - 20}- \dfrac{3}{p^2 - p - 90} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{2}{2p + 18} = \dfrac{2}{2(p + 9)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{2}{2p - 20} = \dfrac{2}{2(p - 10)}$ We can factor the quadratic in the third term: $ \dfrac{3}{p^2 - p - 90} = \dfrac{3}{(p + 9)(p - 10)}$ Now we have: $ \dfrac{2}{2(p + 9)}+ \dfrac{2}{2(p - 10)}- \dfrac{3}{(p + 9)(p - 10)} $ The least common multiple of the denominators is: $ 4(p + 9)(p - 10)$ In order to get the first term over $4(p + 9)(p - 10)$ , multiply by $\dfrac{2(p - 10)}{2(p - 10)}$ $ \dfrac{2}{2(p + 9)} \times \dfrac{2(p - 10)}{2(p - 10)} = \dfrac{4(p - 10)}{4(p + 9)(p - 10)} $ In order to get the second term over $4(p + 9)(p - 10)$ , multiply by $\dfrac{2(p + 9)}{2(p + 9)}$ $ \dfrac{2}{2(p - 10)} \times \dfrac{2(p + 9)}{2(p + 9)} = \dfrac{4(p + 9)}{4(p + 9)(p - 10)} $ In order to get the third term over $4(p + 9)(p - 10)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{3}{(p + 9)(p - 10)} \times \dfrac{4}{4} = \dfrac{12}{4(p + 9)(p - 10)} $ Now we have: $ \dfrac{4(p - 10)}{4(p + 9)(p - 10)} + \dfrac{4(p + 9)}{4(p + 9)(p - 10)} - \dfrac{12}{4(p + 9)(p - 10)} $ $ = \dfrac{ 4(p - 10) + 4(p + 9) - 12} {4(p + 9)(p - 10)} $ Expand: $ = \dfrac{4p - 40 + 4p + 36 - 12}{4p^2 - 4p - 360} $ $ = \dfrac{8p - 16}{4p^2 - 4p - 360}$ Simplify: $ = \dfrac{2p - 4}{p^2 - p - 90}$